We consider the problem of sequentially conducting multiple experiments where each experiment corresponds to a hypothesis testing task. At each time point, the experimenter must make an irrevocable decision of whether to reject the null hypothesis (or equivalently claim a discovery) before the next experimental result arrives. The goal is to maximize the number of discoveries while maintaining a low error rate at all time points measured by Local False Discovery Rate (LFDR). We formulate the problem as an online knapsack problem with exogenous random budget replenishment. We start with general arrival distributions and show that a simple policy achieves a $O(\sqrt{T})$ regret. We complement the result by showing that such regret rate is in general not improvable. We then shift our focus to discrete arrival distributions. We find that many existing re-solving heuristics in the online resource allocation literature, albeit achieve bounded loss in canonical settings, may incur a $\Omega(\sqrt{T})$ or even a $\Omega(T)$ regret. With the observation that canonical policies tend to be too optimistic and over claim discoveries, we propose a novel policy that incorporates budget safety buffers. It turns out that a little more safety can greatly enhance efficiency -- small additional logarithmic buffers suffice to reduce the regret from $\Omega(\sqrt{T})$ or even $\Omega(T)$ to $O(\ln^2 T)$. From a practical perspective, we extend the policy to the scenario with continuous arrival distributions, time-dependent information structures, as well as unknown $T$. We conduct both synthetic experiments and empirical applications on a time series data from New York City taxi passengers to validate the performance of our proposed policies. Our results emphasize how effective policies should be designed in online resource allocation problems with exogenous budget replenishment.

翻译：本文研究顺序进行多重实验的问题，其中每个实验对应一个假设检验任务。在每个时间点，实验者必须在下一个实验结果到达前，对是否拒绝原假设（即宣称发现）做出不可撤销的决策。目标是在所有时间点上最大化发现数量，同时通过局部错误发现率（LFDR）保持较低的错误率。我们将该问题建模为具有外生随机预算补充的在线背包问题。首先针对一般到达分布，证明一个简单策略可实现$O(\sqrt{T})$的遗憾界，并通过理论分析表明该遗憾率在一般情况下不可改进。随后将研究重点转向离散到达分布。我们发现在线资源分配文献中许多现有的重求解启发式策略，虽然在典型设定下能实现有界损失，但可能产生$\Omega(\sqrt{T})$甚至$\Omega(T)$的遗憾。通过观察发现典型策略往往过于乐观且过度宣称发现，我们提出一种融入预算安全缓冲的新型策略。研究证明适度增加安全性可显著提升效率——仅需附加对数级缓冲即可将遗憾从$\Omega(\sqrt{T})$乃至$\Omega(T)$降低至$O(\ln^2 T)$。从实践角度，我们将该策略扩展至连续到达分布、时变信息结构以及未知$T$的场景。通过合成实验和基于纽约市出租车乘客时间序列数据的实证应用，验证了所提策略的性能。我们的研究结果揭示了在外生预算补充的在线资源分配问题中应如何设计有效策略。